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MODULAR FORMS-page92 - 88 LECTURE 8 THE MODULAR CURVE 8.6...

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88 LECTURE 8. THE MODULAR CURVE 8.6 Finally we interpret the spaces M k (Γ) as the spaces L ( D ) for some D on the Riemann surface X (Γ). To state it in a convenient form let us generalize divisors to admit rational coefficients. We define a Q -divisor as a function D : X Q with a finite support. We continue to write D as a formal linear combination D = P a x x of points x X with rational coefficients a x . The set of Q -divisors form an abelian group which we shall denote by Div( X ) Q . For any x Q we denote by x the largest integer less or equal than x . For any Q -divisor D = P a x x we set D = X a x x. Theorem 8.7. Let D = 1 2 r 2 X i =1 x i + 2 3 r 3 X i = r 1 +1 x i + r X i =1 c i , D c = D - 1 k r X i =1 c i , where x 1 , . . . , x r 1 are elliptic points of order 2, x r 1 +1 , . . . , x r 1 2+ r 2 are elliptic points of order 3 and c 1 , . . . , r are cusps. There is a canonical isomorphism of vector spaces M k (Γ) = L ( kK X + kD )) , M k (Γ) 0 = L ( kK X + kD c ) .
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